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\begin{document}
\title{6320-001 - Spring 2021 - Week 10 (3/23, 3/25)}
\maketitle
\section{This week in 6320}
{\bf 3/23} -- More on algebraic closures; finite fields.
{\bf 3/25} -- Separability, field automorphisms, Galois extensions.
\section{Comments and suggested reading}
Dummit and Foote, 13.4, 13.5, 14.1, 14.3
\section{Preview of next week}
\noindent{\it (This is subject to change depending on how far we get this week!)}
\noindent
The fundamental theorem of Galois theory (14.2), cyclotomic fields (parts of 13.3, 13.6, 14.5).
\section{Homework}
\noindent {\bf Due Tuesday, \emph{March 30}, at 11:59pm on Gradescope}
\noindent \emph{All solutions must be typeset using TeX and submitted via Gradescope; handwritten or late submissions will not be accepted. All exercises and problems submitted must start with the statement of the exercise or problem.} \\
\noindent You may work in groups, but you must write up your final solutions individually. Any instances of academic misconduct will be taken very seriously.\\
\noindent\emph{Justify your answers carefully!}\\
\subsection{Exercises} {\it Complete and turn in ALL exercises}: \hfill\\
\noindent Grading scale (for each part of an exercise): \\
3 points -- A correct, clearly written solution\\
2 points -- Right idea, but a minor mistake or not clearly argued\\
1 point -- Some progress but multiple minor mistakes or a major mistake\\
0 points -- Nothing written, totally incorrect, or no substantive progress made towards a solution.\\
\noindent{\bf Exercise 1.} ({\bf DF} 14.3- Exercise 1).
Factor $x^8-x$ into irreducibles in $\mbb{Z}[x]$ and $\mbb{F}_2[x].$
\hfill\\
\noindent{\bf Exercise 2.}
\begin{enumerate}
\item Find two distinct irreducible degree 3 monic polynomials $f(x)$ and $g(x)$ in $\mbb{F}_3[x].$
\item For your $f$ and $g$ as in (1), exhibit an explicit isomorphism
\[ \mbb{F}_3[x]/(f(x)) \xrightarrow{\sim} \mbb{F}_3[x]/(g(x)). \]
\end{enumerate}
\hfill\\
\noindent{\bf Exercise 3.}
Let $K$ be a field of characteristic $p$, and let $a \in K$.
\begin{enumerate}
\item Show $f(x)=x^p - x - a$ is separable.
\item Show that if $\alpha$ is a root of $f(x)=x^p-x-a$, and $k \in \mbb{F}_p$, then $\alpha+k$ is also a root of $f(x)$.
\item If $f$ is irreducible, deduce that $L=K[x]/f(x)$ is a splitting field of $f$, and show that $\mr{Aut}(L/K)\cong \mbb{F}_p$ (as a group under addition).
\end{enumerate}
\hfill\\
\noindent{\bf Exercise 4.}
\begin{enumerate}
\item For $k$ a field, and $a, b, c, d \in k$ such that $ad-bc \neq 0$, show that there exists a unique automorphism of $k(t)$ fixing $k$ and sending $t$ to $\frac{at + b}{ct+d}$.
\item Show that the there is a group homomorphism $\mr{PGL}_2(k) \hookrightarrow \mr{Aut}(k(t)/k)$ sending the equivalence class of
\[ \begin{bmatrix} a & c \\ b & d \end{bmatrix} \]
to the automorphism described in part (1).
\item Give a simple description of the fixed field of the automorphism $t \mapsto t + 1$ of $k(t)$. {\bf Hint: your answer will depend on the characteristic.}
\end{enumerate}
\hfill\\
\noindent{\bf Exercise 5.} Let $p$ be a prime number, let $\overline{\mbb{F}_p}$ be an algebraic closure of $\mbb{F}_p$, let $K= \overline{\mbb{F}_p}(s,t) \; ( =\mr{Frac} (\overline{\mbb{F}_p}[s,t]) )$ and let $L=K(s^{1/p},t^{1/p})$ (by which we mean any splitting field of $(x^p-s)(x^p-t)$).
\begin{enumerate}
\item Show that $[L:K]=p^2$.
\item Show that $|\mr{Aut}(L/K)|=1.$
\item Show there are infinitely many intermediate field extensions
\[ K \subsetneq M \subsetneq L. \]
\end{enumerate}
\hfill\\
\subsection{Problems} {\it Attempt as many as you have time for, but only turn in one (of your choice).} \hfill\\
{\bf Grading scale }(for the problem you turn in): \\
10 points - A correct, complete, and clearly written solution. \\
8 points - Right idea, but one or two minor mistakes or not clearly argued.\\
5 points - Some progress but several minor mistakes or a major mistake. \\
0 points - Nothing written, totally incorrect, or no substantive progress made.\\
{\bf Revision policy: }{\it If you score at least 5 points on the problem you turn in then you will be allowed to submit {\bf one} revision to your solution before the final exam (May 3, 10:30am). If the revision is correct, complete, and clearly written then your mark will change to 9 points. This policy only applies to the problem you submit, not to the exercises in the previous section. }
\hfill\\
\noindent {\bf Problem 1} A field extension $L/K$ is \emph{simple} if there exists $\alpha \in L$ such that $L=K(\alpha)$. Such an $\alpha$ is called a \emph{primitive element} for $L/K$. In this problem, you will prove\\
\noindent{\bf The Primitive Element Theorem:} A finite extension $L/K$ is simple if and only if it admits only finitely many intermediary extensions $K \subseteq M \subseteq L.$\\
{\bf\noindent Remarks:}
Note that Exercise 5 gave an example of a finite extension with infinitely many subextensions; the primitive element theorem then implies that this is also an example of a finite extension that is not simple. On the other hand, the fundamental theorem of Galois theory will imply that for any finite \emph{separable} extension, there are only finitely many subextensions. Thus, as a consequence, any finite extension of a perfect field (in particular, a characteristic zero field) is simple.
Proof of the primitive element theorem:
\begin{enumerate}
\item Show that if a finite extension $L/K$ admits a primitive element (i.e. $L=K(\alpha)$ for some $\alpha \in L$), then there are only finitely many intermediary extensions. {\bf Hint: for $K \subseteq M \subseteq L$, show that the coefficients of the minimal polynomial of $\alpha$ over $M$ generate $M/K$.}
\item Show that if $K$ is a finite field, then \emph{every} finite extension $L/K$ is simple and admits only finitely many intermediary extensions.
\item Show that if $L/K$ is a finite extension with only finitely many intermediary extensions, $K$ is infinite, and $\alpha, \beta \in L$, then there exists $k \in K$ such that $K(\alpha, \beta)=K(\alpha + k \beta).$
\item Conclude.
\end{enumerate}
\end{document}